"Please Please Me" was not only a song title, but also the name of The Beatles' first LP in the UK. It was recorded and released in January of 1963 to capitalize on the success of this song, which had burned up the British charts. There were 14 tunes on the "Please Please Me" album, and eight of them were written by The Beatles, including this one. It was also the second single ever released by The Beatles following "Love Me Do." According to the WikiWackyWoo:

"Please Please Me" has a diverse history. George Martin has stated
that the original version of this song was "rather dreary", was too slow
and consequently had little prospect of being the big hit the band were
looking for. Martin said, "I was still thinking that we should release
their [earlier] recording of "How Do You Do It?"", a previously taped Mitch Murray
composition that Martin insisted the Beatles record which he had
seriously considered as an alternative debut single instead of "Love Me
Do". The group replied that they were only interested in recording their
own material.[7]
McCartney said: "It was symptomatic of our group that we turned down
"How Do You Do It?". Ringo Starr commented: "I remember us all being
ready to stand up for the principle of, 'We have written these songs and
we want to do them'". George Martin was ultimately sympathetic to their
appeals, but said later: "[I] would still have issued "How Do You Do
It?" had they not persuaded me to listen to another version of "Please
Please Me".[11]Lennon first conceived "Please Please Me" as a bluesy, slow tempo
song. Lennon recalled: "I remember the day I wrote it, I heard Roy Orbison doing "Only the Lonely", or something. And I was also always intrigued by the words to a Bing Crosby
song that went, 'Please lend a little ear to my pleas'. The double use
of the word 'please'. So it was a combination of Roy Orbison and Bing
Crosby".[7] Originally it was vocally sparse, did not contain any harmonies or responses, nor did it have the scaled harmonica intro.

This tune was released at the end of November of 1963 and -- had it not been for "She Loves You" -- would have gone right to #1 on the British charts. It had to settle for the #2 spot on its release date. According to the Wiki:

It was also the group's first American number one, entering the Billboard Hot 100 chart on 13 January 1964 at number forty-five and starting the British invasion
of the American music industry. By 1 February it held the number-one
spot, and stayed there for seven weeks before being replaced by "She
Loves You", a reverse scenario of what had occurred in Britain. It
remained on the US charts for a total of fifteen weeks.[6] "I Want to Hold Your Hand" became the Beatles' best-selling single worldwide.[7] In 2013, Billboard magazine named it the 44th biggest hit of "all-time" on the Billboard Hot 100 charts.[8]

This was The Beatles' first tune to be recorded on 4-track and it took 17 takes to get a version they were happy with.

In 2000, McCartney said the initial idea for the song began with Bobby Rydell's hit "Forget Him" with its call and response pattern,
and that "as often happens, you think of one song when you write
another ... I'd planned an 'answering song' where a couple of us would
sing 'she loves you' and the other ones would answer 'yeah yeah'. We
decided that was a crummy idea but at least we then had the idea of a
song called 'She Loves You'. So we sat in the hotel bedroom for a few
hours and wrote it—John and I, sitting on twin beds with guitars." It
was completed the following day at McCartney's family home in Forthlin
Road, Liverpool.[4]

On that same Rolling Stone list of the 500 Greatest Songs of All Time, "She Loves You" is ranked at #64 and is the best selling Beatles' single in history.

Twist And Shout

This is the only song on this list not written by Lennon-McCartney. The Wiki fills in the rest of the story:

This rollicking rocker was recorded at the famous recording session of February 11, 1963, during which The Beatles recorded 10 complete tunes in 13 hours. They left it as the last song to be recorded as they knew it would tear up John's throat. They were right. The take we hear is the first one. They tried for a second, but John had no voice left.

Can't Buy Me Love

This tune was recorded partially in Paris and then finished at the famed Abbey Road studios. Once again, we go to the WikiWackyWoo:

"Can't Buy Me Love" was recorded on 29 January 1964 at EMI's Pathe Marconi Studios in Paris, France,[10] where the Beatles were performing 18 days of concerts at the Olympia Theatre. At this time, EMI's West Germany branch, Odeon, insisted that the Beatles would not sell records in any significant numbers in Germany unless they were actually sung in the German language[11] and the Beatles reluctantly agreed to re-record the vocals to "She Loves You" and "I Want to Hold Your Hand"
prior to them being released in Germany. George Martin travelled to
Paris with a newly mastered rhythm track for what was to be "Komm, Gib Mir Deine Hand" ("Come, Give Me your Hand"/"I Want to Hold your Hand"). "Sie Liebt Dich" ("She Loves You") required the Beatles to record a new rhythm track as the original two-track recording had been scrapped.[10]
EMI sent a translator to be present for this recording session which
had been hurriedly arranged to tie in with the Beatles' Paris
commitments. This was accomplished well within the allotted studio time
allowing the Beatles an opportunity to record the backing track, with a
guide vocal, to the recently composed "Can't Buy Me Love".[11][10]
At this stage the song included background vocal harmonies. But after
listening to the first take, the band concluded that the song did not
need them. Therefore, "Can't Buy Me Love" became the first single the
Beatles released without their characteristic background harmonies.
McCartney's final vocal was overdubbed at EMI Studios, Abbey Road,
London, on 25 February.[6] Also re-recorded on this day at EMI Studios was George Harrison's
modified guitar solo, although his original solo can still just be
heard in the background. Harrison said: "What happened was, we recorded
first in Paris and re-recorded in England. Obviously they'd tried to
overdub it, but in those days they only had two tracks, so you can hear
the version we put on in London, and in the background you can hear a
quieter one."[12]Helen Shapiro, a friend of the Beatles and present at this overdub session, says that Ringo Starr also added extra cymbals "over the top" and that "apparently this was something he did quite often on their records"[13] (Geoff Emerick, tape operator and later the Beatles' recording engineer, credits Norman Smith, the Beatles' then-current engineer, with this overdub).[14]
"Can't Buy Me Love" is also the only English-language Beatles track
that the Beatles themselves recorded in a studio outside the UK,
although the instrumental portion of the Beatles' B-side "The Inner Light" was recorded in India by Indian session musicians.

This song was written in Paris just prior to the sessions and made #295 on the Rolling Stone 500 Greatest Hits of All Time list.

• Until Billboard began using SoundScan for their charts in 1991, it had the biggest jump to number one: (number twenty-seven to number one; no other single had ever done this).

• It gave the Beatles three consecutive number-one songs ("I Want to Hold Your Hand" was replaced at number one by "She Loves You" which was in turn replaced by "Can't Buy Me Love"). The three songs spent a combined total of 14 consecutive weeks at No. 1. This is the only time an artist had three number ones in a row. The best prior was Elvis, who had two consecutive number ones with Don't Be Cruel and Love Me Tender.[16]